find the equation of the plane passing through the points (0,0,0) and (3,-1,2) and parallel to the line x-4/1=y+3/-4=z+1/7.
 

Asked by Shiva Raman | 12th Nov, 2014, 09:32: PM

Expert Answer:

T h e space e q u a t i o n space o f space a n y space p l a n e space p a s sin g space t h r o u g h space t h e space o r i g i n space i s a x plus b y plus c z equals 0... left parenthesis 1 right parenthesis H e r e comma space a comma b space a n d space c space a r e space t h e space d i r e c t i o n space r a t i o s space o f space t h e space n o r m a l space t o space t h e space p l a n e. S i n c e space t h e space p l a n e space i s space p a r a l l e l space t o space t h e space g i v e n space l i n e comma space fraction numerator x minus 4 over denominator 1 end fraction equals fraction numerator y plus 3 over denominator minus 4 end fraction equals fraction numerator z plus 1 over denominator 7 end fraction comma space t h e space d i r e c t i o n space r a t i o s space o f space t h e space p l a n e a r e space p e r p e n d i c u l a r space t o space t h e space l i n e. T h e r e f o r e comma space w e space h a v e comma a open parentheses 1 close parentheses plus b open parentheses minus 4 close parentheses plus c open parentheses 7 close parentheses equals 0 rightwards double arrow a minus 4 b plus 7 c equals 0.... left parenthesis 2 right parenthesis S i n c e space t h e space p l a n e space a l s o space p a s s e s space t h r o u g h space t h e space p o i n t space open parentheses 3 comma minus 1 comma 2 close parentheses comma space w e space h a v e comma a open parentheses 3 close parentheses plus b open parentheses minus 1 close parentheses plus 2 open parentheses c close parentheses equals 0 rightwards double arrow 3 a minus b plus 2 c equals 0.... left parenthesis 3 right parenthesis S o l v i n g space t h e space e q u a t i o n s space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space h a v e comma fraction numerator a over denominator minus 8 plus 7 end fraction equals fraction numerator b over denominator 21 minus 2 end fraction equals fraction numerator c over denominator minus 1 plus 12 end fraction rightwards double arrow fraction numerator a over denominator minus 1 end fraction equals b over 19 equals c over 11 T h u s comma space t h e space e q u a t i o n space o f space t h e space r e q u i r e d space p l a n e space i s minus x plus 19 y plus 11 z equals 0

Answered by Vimala Ramamurthy | 13th Nov, 2014, 09:18: AM